On the set of non radiative solutions for the energy critical wave equation
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| Format: | Preprint |
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2024
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| _version_ | 1866910819550232576 |
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| author | Côte, Raphaël Laurent, Camille |
| author_facet | Côte, Raphaël Laurent, Camille |
| contents | Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$:\[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \]for some $R \> 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein.We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_14932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the set of non radiative solutions for the energy critical wave equation Côte, Raphaël Laurent, Camille Analysis of PDEs Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$:\[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \]for some $R \> 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein.We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field. |
| title | On the set of non radiative solutions for the energy critical wave equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.14932 |